Problem 115 - Chemistry The Molecular Science

Question

You and your lab partner are each given metal cubes that look similar. Your assignment is to make length and mass measurements and use only these data to determine whether the metal is the same in each cube. Your cube is \(1.32 \mathrm{~cm}\) on each edge and has a mass of \(16.23 \mathrm{~g}\). Your partner's cube has a mass of \(24.64 \mathrm{~g}\) and each edge measures \(1.46 \mathrm{~cm}\). Your partner says that the metal is the same in each cube; you don't agree. Refute your partner's conclusion.

Step-by-Step Solution

Answer

The cubes are made of different metals; their densities are different: 7.06 g/cm³ vs. 7.92 g/cm³.

Step 1: Calculate Volume of Your Cube

The formula for the volume of a cube is \( V = ext{side}^3 \). For your cube, each side is \( 1.32 \text{ cm} \). Therefore, the volume is \( V = 1.32^3 \text{ cm}^3 \). Calculating this, \( V = 2.30 \text{ cm}^3 \).

Step 2: Calculate Density of Your Cube

Density is given by the formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). For your cube, the mass is \( 16.23 \text{ g} \), and the volume is \( 2.30 \text{ cm}^3 \). So, the density is \( \frac{16.23}{2.30} = 7.06 \text{ g/cm}^3 \).

Step 3: Calculate Volume of Your Partner's Cube

Using the formula for the volume of a cube again, for your partner's cube with a side of \( 1.46 \text{ cm} \), the volume is \( 1.46^3 \text{ cm}^3 \). Calculating this, \( V = 3.11 \text{ cm}^3 \).

Step 4: Calculate Density of Your Partner's Cube

Similarly, calculate the density of your partner's cube. The mass is \( 24.64 \text{ g} \) and the volume is \( 3.11 \text{ cm}^3 \). Thus, the density is \( \frac{24.64}{3.11} = 7.92 \text{ g/cm}^3 \).

Step 5: Compare Densities

Your cube has a density of \( 7.06 \text{ g/cm}^3 \) and your partner's cube has a density of \( 7.92 \text{ g/cm}^3 \). Since the densities are different, the metal in each cube cannot be the same.

Key Concepts

Volume Calculation

Volume is a measure of the amount of space an object occupies. Calculating the volume of a cube is straightforward because all sides are equal in length. To compute the volume, use the formula:\[ V = ext{side}^3 \] This means you multiply the length of one side by itself twice. It's quite simple as it only requires knowing the length of one edge. For example, if a cube has a side that measures 1.32 cm, as in our exercise, the volume is calculated by raising this length to the power of three:- Your cube's volume: \( V = 1.32^3 \) - Performing the calculation results in a volume of 2.30 cm³.For your partner's cube, the process is identical:- Partner's cube's volume: \( V = 1.46^3 \) - The volume, when calculated, is 3.11 cm³.Both of these measures provide a vital piece of the puzzle when determining the material's identity.

Mass Measurement

Mass is the amount of matter in an object and is a crucial component in density determination. In this exercise, measuring the mass accurately is essential since it's used to derive the material density. Both you and your partner were given cubes with specific masses: - Your cube’s mass: 16.23 g - Partner's cube’s mass: 24.64 g When measuring mass, - Ensure the scales are calibrated correctly - Measure in a unit consistent with volume for ease of calculation (grams, in this case) Precision is necessary, as even slight variations in mass can affect the calculated density, thereby influencing the determination of the material type.

Material Identification

Identifying the material of an object based on measurements often involves calculating its density. Density is defined as mass per unit volume and is a distinguishing property for materials. Here's how you calculate density:\[ ext{Density} = \frac{ ext{Mass}}{ ext{Volume}} \]For your cube, given a mass of 16.23 g and a volume of 2.30 cm³, the density is:- Your density: \( \frac{16.23}{2.30} \approx 7.06 \text{ g/cm}^3 \)Your partner’s cube, with a mass of 24.64 g and a volume of 3.11 cm³, yields:- Partner's density: \( \frac{24.64}{3.11} \approx 7.92 \text{ g/cm}^3 \)By examining the densities, it's clear that they do not match:- Your cube: 7.06 g/cm³- Partner's cube: 7.92 g/cm³This significant difference indicates that the cubes are made of different materials, refuting your partner's conclusion of similarity.